BOUNDS FOR TOEPLITZ DETERMINANTS OF A CERTAIN SET OF ANALYTIC FUNCTIONS DEFINED BY -DIFFERENTIATION AND HYPERBOLIC COSINE FUNCTION

Abstract

In complex analysis, determinants are used to study properties of matrices arising from complex-valued functions, particularly in transformations, residue computations, Jacobians for conformal mappings, systems of complex equations, and in evaluating special matrices like Toeplitz and Hankel. Thus, this study is on a certain set  consisting of analytic and univalent functions of the Taylor’s series  defined by using and maps the unit disk  onto a domain defined by the hyperbolic cosine function  In the methodology, the principles of quantum derivative (q-derivative) operator, subordination, q-series expansion, and some widely acknowledged lemmas are adopted. The established results include the initial coefficient bounds for  and Toeplitz determinants: , , , and . These results however generalised many existing ones thereby expanding the scope of its applications in areas such as in the solution to analytical problems of orthogonal polynomials, determinants, and frequency analysis where the special cases of the Toeplitz determinants are involved.